In the implementation of these methods for practical applications, perhaps the most subtle part is the phase adjustment due to encounters with caustics, where trajectories (or rays) pass through singularities in the prefactor that weights the contributions, i.e. the inverse square root of a determinant that vanishes. In the context of quantum mechanics, where the phase change is governed by what is typically called the Maslov index, a great deal has been written on this subject. For example, there are notable works by Keller [1]}, Maslov and Fedoriuk [2]}, Gutzwiller [3]}, and Littlejohn and co-authors [4]}. In the context of the stationary phase approximation applied to Feynman path integrals, which in its simplest form generates the Van Vleck-Gutzwiller propagator [5]} approximation to the time-dependent Green function for the Schrödinger equation, Schulman connects the index to Morse theory [6]}. To paraphrase Keller [1]}: `the prefactor's phase is retarded by \(m\pi /2\)  on a trajectory which passes through a caustic on which the determinant vanishes to the \(m^{th}\)  order.' The Maslov index, \(\nu \) , is therefore just the sum of these m-integers for all of the caustics a trajectory passes through in its full history. The end result is that the absolute value of the prefactor's inverse square root of a determinant gets multiplied by \(\exp (-i\nu \pi /2)\) , and the index can be considered to be defined modulus four [8]}.
